Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C25 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C22⋊C4×C25 |
| unambiguous semidirect product | C22⋊(C42×C25) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C25 | C22 |
| commutator | C2 | C2×C42×C25 |
| Frattini, socle, cosocle | C23×C5 | C23×C5 |
| prime radical | C24×C5 | C22×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C25 ◁ G | |
387 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 387 conjugacy classes of subgroups — too many to draw. Only its 219 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(5,6,8,10)(7,9,11,12);(6,9)(10,12)1000 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 878);